We concern with a vector-borne disease model with horizontal transmission and infection age in the host population. With the approach of Lyapunov functionals, we establish a threshold dynamics, which is completely determined by the basic reproduction number. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable while if the basic reproduction number is larger than one then the infected equilibrium attracts all solutions with initial infection. These theoretical results are illustrated with numerical simulations.

Figure 1.
When $R_0<1$, the infection-free equilibrium $E^0$ of (2) is globally asymptotically stable. Here since $E_h(t)$ converges to $0$ very fast, we use the time interval $[0, 100]$ different from the interval $[0, 1000]$ for other components